Question:
The integral \( \int \frac{dx}{1 + e^x} \) is:
The integral \( \int \frac{dx}{1 + e^x} \) is:
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When solving integrals of rational functions, recognize standard forms for the integrals. The integral \( \int \frac{dx}{1 + e^x} \) directly leads to the logarithmic form \( \log|1 + e^x| + C \).
Updated On: Apr 7, 2026
- \( e^x + C \)
- \( \log|1 + e^x| + C \)
- \( \log|1 + e^{-x}| + C \)
- \( \log |1 - e^{-x}| + C \)
- \( \log |1 - e^x| + C \)
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The Correct Option is B
Solution and Explanation
We are tasked with solving the integral: \[ \int \frac{dx}{1 + e^x} \] This integral is of a standard form. We recognize that the integral of \( \frac{1}{1 + e^x} \) is directly related to the natural logarithm function.
Specifically: \[ \int \frac{dx}{1 + e^x} = \log|1 + e^x| + C \] where \( C \) is the constant of integration. Thus, the correct answer is \( \log|1 + e^x| + C \).
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